Assume we have two sequences of rv. defined on the same probability space $\{X_{n}\}$ and $\{Y_{n}\}$ such that $$ X_{n} \stackrel{d}{\to} X $$
$$ Y_{n} \stackrel{d}{\to} Y $$ and $P[X_{n} \geq Y_{n}] \to 1$ as $n\to\infty$. Then what can be said about $X$ and $Y$? Is it true that $X\geq_{s}Y$? Here $\geq_{s}$ means stochastically greater.
Assume for simplicity of notations that $X$ and $Y$ are defined on the same probability space. Let $t$ be a continuity point of the cumulative distribution function of $X$ and $Y$. Then \begin{align} \mathbb P \left\{Y \gt t\right\} &= \lim_{n\to+ \infty } \mathbb P \left\{Y_n \gt t\right\} \\ & \leqslant \limsup_{n\to+ \infty } \mathbb P \left\{X_n \lt Y_n\right\} + \mathbb P \left( \left\{Y_n \gt t\right\} \cap \left\{X_n \geqslant Y_n\right\}\right)\\ & \leqslant \limsup_{n\to+ \infty } \mathbb P \left\{X_n \lt Y_n\right\} + \limsup_{n\to+ \infty } \mathbb P \left\{X_n \gt t\right\} \\ &=\mathbb P \left\{X \gt t\right\}. \end{align}
Since the functions $g_1\colon t\mapsto \mathbb P \left\{X \gt t\right\}$ and $g_1\colon t\mapsto \mathbb P \left\{Y \gt t\right\}$ are right-continuous and the discontinuity points of $g_1-g_2$ is at most countable, the equality $$ \mathbb P \left\{Y \gt t\right\} \leqslant \mathbb P \left\{X \gt t\right\} $$ holds for any real number $t$.